08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access An approximation of solutions to heat equations defined by generalized measure theoretic Laplacians(2020) Ehnes, Tim; Hambly, BenWe consider the heat equation defined by a generalized measure theoretic Laplacian on [0, 1]. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure μ, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.Item Open Access Analysis of target data-dependent greedy kernel algorithms : convergence rates for f-, f· P- and f/P-greedy(2022) Wenzel, Tizian; Santin, Gabriele; Haasdonk, BernardData-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.Item Open Access Ordinal patterns in clusters of subsequent extremes of regularly varying time series(2020) Oesting, Marco; Schnurr, AlexanderIn this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns giving the relative positions of the data points within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine.Item Open Access On quiver Grassmannians and orbit closures for gen-finite modules(2021) Pressland, Matthew; Sauter, JuliaWe show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.Item Open Access Lossless transformations and excess risk bounds in statistical inference(2023) Györfi, László; Linder, Tamás; Walk, HarroWe study the excess minimum risk in statistical inference, defined as the difference between the minimum expected loss when estimating a random variable from an observed feature vector and the minimum expected loss when estimating the same random variable from a transformation (statistic) of the feature vector. After characterizing lossless transformations, i.e., transformations for which the excess risk is zero for all loss functions, we construct a partitioning test statistic for the hypothesis that a given transformation is lossless, and we show that for i.i.d. data the test is strongly consistent. More generally, we develop information-theoretic upper bounds on the excess risk that uniformly hold over fairly general classes of loss functions. Based on these bounds, we introduce the notion of a δ -lossless transformation and give sufficient conditions for a given transformation to be universally δ -lossless. Applications to classification, nonparametric regression, portfolio strategies, information bottlenecks, and deep learning are also surveyed.Item Open Access Approximation of a two‐dimensional Gross-Pitaevskii equation with a periodic potential in the tight‐binding limit(2024) Gilg, Steffen; Schneider, GuidoThe Gross-Pitaevskii (GP) equation is a model for the description of the dynamics of Bose-Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the x‐direction and a harmonic oscillator potential in the y‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case.Item Open Access A note on the Fröhlich dynamics in the strong coupling limit(2021) Mitrouskas, DavidWe revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state ϕ0 ⊗ξα, where ϕ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t = O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation.Item Open Access Dominant and global dimension of blocks of quantised Schur algebras(2021) Fang, Ming; Hu, Wei; Koenig, SteffenGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and Sq(n,r) with n⩾r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59-85, 2021).Item Open Access Implications of modeling seasonal differences in the extremal dependence of rainfall maxima(2022) Jurado, Oscar E.; Oesting, Marco; Rust, Henning W.For modeling extreme rainfall, the widely used Brown-Resnick max-stable model extends the concept of the variogram to suit block maxima, allowing the explicit modeling of the extremal dependence shown by the spatial data. This extremal dependence stems from the geometrical characteristics of the observed rainfall, which is associated with different meteorological processes and is usually considered to be constant when designing the model for a study. However, depending on the region, this dependence can change throughout the year, as the prevailing meteorological conditions that drive the rainfall generation process change with the season. Therefore, this study analyzes the impact of the seasonal change in extremal dependence for the modeling of annual block maxima in the Berlin-Brandenburg region. For this study, two seasons were considered as proxies for different dominant meteorological conditions: summer for convective rainfall and winter for frontal/stratiform rainfall. Using maxima from both seasons, we compared the skill of a linear model with spatial covariates (that assumed spatial independence) with the skill of a Brown-Resnick max-stable model. This comparison showed a considerable difference between seasons, with the isotropic Brown-Resnick model showing considerable loss of skill for the winter maxima. We conclude that the assumptions commonly made when using the Brown-Resnick model are appropriate for modeling summer (i.e., convective) events, but further work should be done for modeling other types of precipitation regimes.Item Open Access Symplectic model order reduction with non-orthonormal bases(2019) Buchfink, Patrick; Bhatt, Ashish; Haasdonk, BernardParametric high-fidelity simulations are of interest for a wide range of applications. But the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such a ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.